Pre-routing Estimation of Shielding for RLC Signal Integrity�

JamesD. Z. Ma, Arvind Parihar, andLei HeECEDepartment,Universityof Wisconsin,Madison,WI 53706

Abstract

Theformula-based�������

modelis a figure of merit forthe inductivecoupling,and hasbeenusedto solvethe si-multaneousshield insertion and net ordering (SINO) andsimultaneoussignalandpowerrouting(SPR)problems.Inthis paper, we first showthat the

� �����modelhas a high

fidelity compared to the SPICE-computednoiseunder anaccurateRLCcircuit model.Wethendevelopsimpleyetac-curate formulaeto estimatenumbersof shieldsneededbyoptimal SINOsolutionsunderthe

�������model. Extensive

experimentsshowthat our pre-routing estimationhas er-rors lessthan10%comparedto solutionsgivenbydetailedSINOalgorithms. Theseformulaecan be usedeffectivelyasa pre-routing congestionestimationfor layout planningandsynthesis.

1. Introduction

It is becomingincreasinglyevident that on-chip induc-tance,especiallymutual inductance,shouldbe consideredfor high-performanceinterconnectdesign. Several recentworkhavestudiedinterconnectoptimizationconsideringin-ductive coupling for multiple couplednets. Examplesin-clude the simultaneousshield insertion and net ordering(SINO) problem [4] and simultaneoussignal and powerrouting(SPR)problem[6].

The�������

modelbut not noisevoltageis usedin [4, 6]to representthe inductive coupling. Despitethat the

�������model is far away from accurate,our first contributionin this paperis to show that for optimal SINO solutions,the

�������model hasa high fidelity comparedto SPICE-

computednoiseunderanaccurateRLC circuit model. I.e.,a SINO solution with a higher coupling under the

�������modelalsohasa highernoisevoltage.

To integratethe SINO probleminto the currentdesignflow, it is desiredto have an efficient interconnectconges-tion estimation.Our secondcontribution of this paperis todevelop simpleyet accurateformulaeto estimatenumbers

This researchwaspartiallysupportedby NSFCAREERAwardCCR-0093273andagrantfrom Intel. Addresscommentsto he@ece.wisc.edu.

of shieldsneededby optimal SINO solutionswithout run-ningSINOalgorithms.

The rest of this paperis organizedas follows: Section2 introducespreliminariesandreviews therelatedresearch.Section3 studiesthefidelity of the

� �����model. Section4

derivesformulaefor thepre-routingcongestionestimation.Section5 concludesthepaperanddiscussesongoingwork.

2. Preliminaries and related work

Similar to [4], we denotea signalnetass-wire andde-fine that two nets �� and � are sensitiveto eachother ifa switchingsignalon �� causes� to malfunction(duetoextraordinarycrosstalkor delayvariation)andvice-versa.In this case, � is an aggressorfor , and a victim of � . The sensitivityrate of �� is definedas the ratio of thenumberof aggressorsfor �� to the total numberof nets.Ashieldis a wire directly connectedto P/Gwires. By insert-ing a shieldbetweentwo sensitive s-wires,we areabletoeliminatethe capacitive couplingandreducethe inductivecoupling.

In orderto efficiently modelinductivecouplingbetweentwo arbitrarys-wires,theinductivecouplingcoefficientbe-tweentwo s-wirescanbe usedasthefigure of merit. Thecouplingcoefficient betweentwo nets � and �� is definedas � � ��� � � �� � ��� � �where � � � is themutualinductancebetween � and �� , and� � and

� � is self inductancefor � and �� , respectively. Leta block denotethe set of wires betweentwo shields. Inthis “effective coupling model” (i.e.,

�������model),when

the two netsare in the sameblock, as shown in Figure 1where � � and ��� aretrackorderingnumbersfor thetwo s-wires,and ��� and �� aretrackorderingnumbersfor thetwoshields,thecouplingcoefficient is computedas

� �"! � �$#&%'#&(")+* � #-,�)�).where %�#"(&) and � #/,�) aretwo functionsdefinedasfollows:assumingthat thefunction % is 0 at � � , andis 1 at � � , %'#&(")is the linear interpolation,i.e., %'#&(") �103254"687:9-;0<2�=>6?7@9-; . Similarly,

� #-,�) � 037�A@6?2 = ;0<7>A:682 4 ; . Whennets ( and , arein differentblocks,the coupling coefficient

� � is 0 or a small constant. Wecomputethetotal inductivecouplingfor � as:

� � �CB�EDF �� � �

for eachs-wire �� thatis sensitive to � .

Figure 1. Illustration of�������

computation.

The� �����

model hasbeenapplied to SINO and SPRproblems[4, 6]. TheSINOproblemperformssimultaneousshieldinsertionandnetordering.TheSPRproblemdefinesan optimal P/G structuresuchthat the minimal numberofshieldsis neededby theoptimalSINOsolutionwith respectto theabove P/Gstructure.Bothproblemsminimizetheto-tal routingarea.Further, resultingsolutionsarefreeof ca-pacitive couplingandsatisfy inductive couplingconstraintunderthe

�������model.

3. Fideltiy of GIH�J�J model

The�������

model is easyto computeandconvenienttouseat a higher designlevel or at an earlierdesignstage.However, it is not clearhow goodthe

�������model is asa

figure of merit for the accuratenoisevoltage. Especially,it assumesthat the currentreturnsfrom the nearestshield,which maynotbetruein general.Thecurrentoftenreturnsfrom quietwireswithin thecurrentblock if thereareplentyof quietwires in this block. On theotherhand,thecurrentoften returnsfrom shieldsor quiet wires outsidethe cur-rentblock whenmultiple wires in thecurrentblock switchsimultaneously. Despitethat the

�������model is far away

from accurate,it hasa high fidelity in the following sense:anoptimalSINO solutionwith a highercouplingvalue

� �alsohasa higherSPICE-computednoiseover a distributedRLC circuit modelusingthepartial inductancemodel.Ourobservationis basedon the following experiment.We firstgeneratea large numberof optimal SINO solutionsusing

SINOalgorithms,andrankthesesolutionsby boththecou-pling value

� � andSPICE-computednoiseusingan accu-rateRLC circuit model.We thencomputetheabsolutedif-ferencebetweenthe two rankingsfor eachoptimal SINOsolution. If the ranking differenceis small enough,the� �����

modelhasa high fidelity. The similar rankingtech-niqueshave beenusedto studythefidelity of Elmoredelaymodelin [1, 2].

Further, we calculatetheSPICE-computednoisediffer-encewith respectto the averagerankingdifferencein thefollowing way: let theaveragerankingdifferencebed. Fora SINO solutionwhoseSPICE-computednoiseranking isi, we computethe relative differencebetweenthe (i+d )-thandi-th SPICE-computednoise,aswell asthatbetweenthe(i-d)-th and i-th SPICE-computednoise. Betweenthe twovalues,theonewith thelargerabsolutevaluedividedby thei-th SPICE-computednoisevalueis definedasnoisediffer-encefor the solution. In otherwords,it shows how muchthenoisedifferenceis introducedif we use

� �����modelto

approximatethe SPICE-computednoisevoltageunderanaccurateRLC circuit model.

technology 0.10K �Vdd 1.05Vfrequency 3GHzinput rising time 33L8driverresistance 150Mloadcapacitance 60fFwire width 1.0K �wire thickness 1.1K �wire spacing 0.8K �wire length 2000K ��ON&P

0.5,1.0,1.5,2.0

Table 1. Experiment settings for SPICE simu-lation

In our SPICEsimulation,we generatea RLC segmentfor each100K � wire segment,anda mutualinductancebe-tweenany pair of two segmentseven thoughthey belongto the samewire. We usethe partial inductancemodel in[8, 3], without assumingany currentreturnpath. The pa-rametersfor theSPICEsimulationof thedetailedRLC cir-cuit modelaresummarizedin Table1. We reportin Figure2 thedistributionof therankingdifferencesfor SINO solu-tions. Theaveragerankingdifferenceis only 124.533over3,840SINO solutions. The averagenoisedifferencewithrespecttosucharankingdifferenceis 6.2018%.In addition,wepresentin Table2 theaveragenoisedifferencefor SINOsolutionswith differentcouplingvalues.Onecaneasilyseethat thesmallerthecoupling,thesmallertheaveragenoisedifference. We assumein our experimentthat the largest

couplingvalueis 2.0 underthe�������

model. This valueischosenbasedon our experiments[4] that for 0.10K � tech-nology,

� N&P � .RQTSis a “good” noiseboundin the sense

thattheresultingSINOsolutionshavetheSPICE-computedmaximumnoisecloseto but still lessthan15%of Vdd.

Figure 2. The distribution of the ranking dif-ference for 3,840 SINO solutions.

couplingvalue noisedifference0 U � � U 0.5 5.1193%

0.5 U � � U 1.0 5.9872%1.0 U � � U 1.5 6.5520%1.5 U � � U 2.0 7.1487%

Table 2. The average noise difference for dif-ferent coupling values

Basedontheaboveempiricalevidence,weconcludethatthe

�������modelhasa high fidelity over SPICE-computed

noiseusingan accuratecircuit model. This fidelity holdsfor optimal SINO solutionsunder practical noisebound.WespeculatethattheSINOalgorithmsareableto distributeshieldsandquitewiresevenlybothspatiallyandtemporally,thereforethecurrentmainlyreturnswithin thecurrentblockincludingthenearestshields.This leadsto thefidelity of the� �����

model.

4. Formulae of pre-routing estimation

We first considerthe uniform sensitivity case,whereeachnet hasthe samesensitivity rate V . Further, We de-notethenumberof shieldsas �XW , thenumberof netsas � ,andthecouplingcoefficient thresholdas

� N&P. As shown in

[6], thenumberof shieldsfor afixed� N&P

is

� W � #&Y � � � *ZY ) � V *[#&YE\ � � *]Y�^�) � V*_Ya` � � *bYac (1)

where Y � to Yac are coefficients determinedempirically.However, theabove formuladid notconsider

� N&Pasavari-

able,which limits its application.We proposethat the numberof shieldsis a function of� , V , and

� N&Pas

�dWe� f SRQ gEgih � � N&P � � � V * SRQTj�.�g � � � V * .RkEQTj�l � �ON&P � V f .EmRQTh�l � V * SRQTnEh�g � � N&P � � � Vof SRQTnEn�l � � � Vf .�.pQTh�n � � N&P � V * npk�Q giq � Vof SpQ-k�. � � N&P � �* SRQT.ES�. � � * qpQ gal � �ON&P f jRQT.�h

(2)

This formulashows thatgivenfixedvaluesof � and V , �dWis a linearfunctionof

� N&P.

Formula (2) can be extended to the case with non-uniform sensitivity rates. When the sensitivity rate is notuniform but is V � for s-wire � , the numberof shieldsisgivenby

� W � f SpQ g�gih � �ON"P � 2B � F � V � * SRQTjE.�g � 2B � F � V

�

* .pk�QTj�l �� N"P� � 2B � F � V

� f .�mpQTh�l k� � 2B � F � V �

* SpQTn�h�g � � N"P � 2B � F � V?��fSRQTn�nEl � 2B � F � V?�

f .E.RQTh�n � �ON"P� � 2B � F � V � *nRkEQ gaq k� � 2B � F � V �f SpQ-k�. � �ON&P � � * SpQ .ES�. � � * qpQ gil � �ON&P

f jpQT.�h (3)

We verify formulae(2) and(3) by comparingnumbersof shieldsgivenby theformulaeandthoseobtainedby theSINO program. As shown in Figure3, the differencebe-tween the estimationand the SINO programis less than10%.Notethatwecantunetheconstantin formulae(2) and(3) to obtaina desiredloweror upperboundon thenumberof shieldsrequiredby SINOsolutions.

5. Conclusion and discussion

In this paper, we have shown that the�������

modelhasa high fidelity to SPICE-computednoiseunderanaccurateRLC circuit model. We have developedsimple yet accu-rateformulaeto estimatethenumberof shieldsneededbyoptimal SINO solutionsunderthe

�������model. Extensive

experimentsshow that our pre-routingestimationhaser-rors lessthan10%comparedto solutionsgivenby detailedSINOalgorithms.Suchestimationcanbeusedby intercon-nectplanningandsynthesis.For example,shieldingesti-mationhasbeenusedto developanefficient two-phaseal-gorithmfor theSPRproblem[6]. Further, we have recently

Figure 3. Comparison of � W by the estimationand the SINO- r program.

developedshieldingestimationfor SINOsolutionsunderanexplicit RLC noiseconstraint[5, 7].

The simplestinterconnectstructure,parallel bus struc-ture,is assumedin thiswork. We areincorporatingthepre-routing estimationandSPRformulationto a global routerfor signalandpowernetco-design.EventhoughtheexplicitRLC noisemodelusedin [5, 7] is moreaccurate,we planto first applythe

�������modelto theglobalrouterdueto the

simplicity andhigh fidelity of the�������

model.Thesenewresultswill bemadeavailableathttp://eda.ece.wisc.edu.

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